Dissipative Control of Singular Systems by Proportional Plus Derivative State Feedback

This paper presents a design procedure for linear time invariant systems using output proportional plus derivative feedback. The derivative feedback is shown to improve the controller performance in the presence of parameter variations and disturbance inputs, at the cost of increased noise response. A quadratic performance index, with the addition of terms to penalize disturbance and noise response and eigenvalue and response sensitivity, is used as the basis for a design method. The proposed method is a generalization of one presented previously for the case of perfect state and state derivative feedback. The method is illlustrated on a simple first order example and on the design of a controller for the lateral dynamnics of an L1011 aircraft. The results indicate that improved performance is obtained through the addition of perfect output derivative feedback, however, much of that improvement is lost when approximate differentiation is used.

This paper is concerned with the problem of robust stability and stabilization of singular systems with uncertainties in both the derivative and state matrices. By using a parameter dependent Lyapunov function, we derive the LMI-based sufficient conditions for the stabilization of the singular systems. Secondly, by solving these LMIs, a proportional plus derivative (PD) state feedback controller is designed for the closed-loop systems to be quadratically normal and quadratically stable (QNQS). Finally, the numerical example is given to show the effectiveness of the proposed theorems.

Mixed H2/ H∞ optimization approach to gap control on EDM

The point-to-point control of manipulators having elastic joints is considered. It is shown that a simple PD (proportional plus derivative) controller, similar to that used for rigid robots, suffices to globally stabilize the elastic joint robots about a reference position. A robustness analysis is also given with respect to uncertainties on the robot parameters. The results of numerical simulation tests of a manipulator with three revolute elastic joints are presented. >

Controllability Analysis of Linear Time-Invariant Descriptor Systems∗∗This work was supported by CSIR, New Delhi (Grant No. 25(0195)11/EMRII).

The paper is concerned with the problem of adaptive tracking system control synthesis. It is assumed that a nonlinear, feedback linearizable object dynamics (model structure) is (partially) unknown and some of its nonlinear characteristics can be approximated by a sort of functional approximators. It has been proven that proportional state feedback plus parameter adaptation are able to assure its asymptotic stability. This form of controller permits online compensation of unknown model nonlinearities and exogenous disturbances, which results in satisfactory tracking performance. An interesting feature of the system is that the whole process control is performed without requisite asymptotic convergence of approximator parameters to the postulated ‘true’ values. It has been noticed that the parameters play rather a role of slack variables on which potential errors (that otherwise would affect the state variables) cumulate. The system's performance has been tested via Matlab/Simulink simulations via an example of ship path‐following problem. Copyright © 2009 John Wiley & Sons, Ltd.

The problem considered is that of finding, for a linear time invariant system x/spl dot/(t)=Ax(t)+Bu(t), y(t)=Cx(t), a PD state feedback u(t)=F/sub d/x/spl dot/(t)+F/sub p/x(t)+r(t) such that y(t)=r(t). Also, geometric conditions which guarantee the existence of such a solution are to be found. Existing methods are briefly discussed, then the authors give an alternative solution in terms of operators without using the Brunovsky canonical form. They assume right-invertibility, which is necessary for row-by-row decoupling.

The authors conducted a study of automatic path tracking performance in the automated highway system (AHS) public trials that were carried out in Japan in 1996. A comparison was made of proportional-plus-derivative (PD) control and linear quadratic (LQ) control as two potential control systems. Because the PD control system lacked state feedback of yaw motion, the interference of yaw motion on lateral motion could not be eliminated. This made it impossible to achieve both satisfactory response and convergence. The LQ control system, on the other hand, had a larger number of state feedback coefficients, enabling lateral motion to be separated from yaw motion. This made it possible to set the response and convergence of lateral motion at will. Driving on an automated highway system is premised on the receipt of feedforward information from the road-side infrastructure. In this study, it was shown both theoretically and experimentally that road curvature can be estimated logically as a vehicle travels along even in the absence of feedforward information on curvature. Logical estimates are obtained with an algorithm in which curvature is treated as a state variable. It was also shown experimentally that an autonomous lane-tracking vehicle can even navigate road curves of unknown curvature by using a regulator configured around the method of estimating road curvature logically.

This thesis is concerned with the dissipative control and filtering problems of singular systems. Four classes of singular systems are considered: delay-free singular systems, singular systems with constant time-delay, uncertain singular systems with time-varying delay and sensor failures, and singular Markovian jump systems with actuator failures.
\nFor delay-free singular systems, the system augmentation approach is employed to study the dissipative control and filtering problems. First, the approach is used to solve the dissipative control problem by static output-feedback for standard state-space systems which are the special cases of singular systems. For a continuous-time standard state-space system, the closed-loop system is represented in an augmented system form. Based on the augmented system, a necessary and sufficient dissipativity condition is proposed, which decouples the Lyapunov matrix and controller matrix. To further separate the Lyapunov matrix and the system matrices, an equivalent condition is obtained by introducing some slack matrices. Then, a necessary and sufficient condition for the existence of a static output-feedback controller is proposed, and an iterative algorithm is given to solve the condition. For discrete-time singular systems, by giving an equivalent representation of the solution set, a necessary and sufficient dissipativity condition is proposed in terms of strict linear matrix inequality (LMI) which can be easily solved by standard commercial software. Then a state-feedback controller design method is given based on the augmentation system approach. The method is extended to the static output-feedback control problem and the reduced-order dissipative filtering problem.
\nFor continuous-time singular time-delay systems, the problem of state-feedback dissipative control is considered. An improved delay-dependent dissipativity condition in terms of LMIs is established by employing the delay-partitioning technique, which guarantees a singular system to be admissible and dissipative. Based on this, a delay-dependent sufficient condition for the existence of a state-feedback controller is proposed to guarantee the admissibility and dissipativity of the closed-loop system. In addition to delay-dependence, the obtained results are also dependent on the level of dissipativity. Moreover, the results obtained unify existing results on H∞ performance analysis and passivity analysis for singular systems.
\nFor discrete-time singular systems with polytopic uncertainties, time-varying delay and sensor failures, the problem of robust reliable dissipative filtering is considered. The filter is designed by the reciprocally convex approach such that the filtering error singular system is admissible and strictly (Q, S, R)-dissipative. For singular systems with time-varying delay and sensor failures, a sufficient condition of reliable dissipative analysis is obtained in terms of LMIs. Then the result is extended to the uncertain case by introducing some variables to decouple the Lyapunov matrices and the filtering error system matrices. Moreover, a desired filter for uncertain singular systems with time-varying delay and sensor failures is obtained by solving a set of LMIs.
\nFor continuous-time singular Markovian jump systems with actuator failures, the problem of reliable dissipative control is addressed. Attention is focused on the state-feedback controller design method such that the closed-loop system is admissible and strictly (Q, S, R)-dissipative. A sufficient condition is obtained in terms of strict LMIs. Moreover, the results obtained unify existing results on H∞control and passive control on singular Markovian jump systems.

Impulsive phenomenon and state jumps at switching instants are inevitable control difficulties in the study of stability for switched singular systems. Proportional-derivative (P-D) state feedback may be an effective way to eliminate impulsive behaviors and state jumps. In this work, the problem of stabilization is studied for switched singular systems in the continuous-time case and discrete-time case. A synchronous design method of P-D state feedback controllers is proposed by introducing some free-weighting matrices. Based on P-D state feedback, some sufficient conditions, which can guarantee that the closed-loop systems are normal and stable (NS), are obtained by using multiple Lyapunov functions. Compared with step-by-step design, synchronous design brings more freedom to the design of P-D state feedback controllers and can better improve the dynamic performance of the systems. Finally, simulation examples are given to demonstrate the effectiveness of the proposed methods.

We study the disturbance decoupling problem for linear time invariant descriptor systems. We give necessary and sufficient conditions for the existence of a solution to the disturbance decoupling problem with or without stability via a proportional and/or derivative feedback that also makes the resulting closed-loop system regular and/or of index at most one. All results are proved constructively based on condensed forms that can be computed using orthogonal matrix transformations, i.e., transformations that can be implemented in a numerically stable way.

Stabilization of a CSTR with two arbitrarily switching modes using modal state feedback linearization

Model matching of descriptor systems by proportional state feedback

Model Matching of Descriptor Systems by Proportional State Feedback

A closed-form solution is presented to the H ∞ sub-optimal control problem via state or output feedback. The problem formulation is general and does not impose any restrictions on feedthrough terms between the control input and the controlled output or between the disturbance input and the measurement output variables. It is, however, required that both the transfer function from the disturbance input to the measurement output and the transfer function from the control input to the controlled output have no invariant zeros on the jw axis and that they satisfy certain geometric conditions. For the same class of systems, the conditions under which the H ∞ optimal control problem, via state feedback, has a solution are given. Moreover, explicit expressions for the optimal solutions are provided. Finally, the pole/zero cancellations in the closed-loop system resulting from the H ∞ optimal or sub-optimal state or output feedback are examined.